I'm working on an actuarial project to estimate monthly probabilities that someone becomes disabled.
A portfolio of persons (each having a different 'PolicyNr') is observed during 12 months, and the time until disability is registered by the variable 'TimetoDisability'. When no disability occured during the 12 months, the variable 'RightCensored' has the value 1. We further have the variables 'Gender', 'AgeatDisability' (which equals the age at the disability, or the age after 12 months for the right censored observations), and the variable 'OccupationClass'.
We have data available in a wide format. For example, the following lines are part of the data:
| PolicyNr | TimetoDisability | RightCensored | Gender | AgeatDisability | OccupationClass |
| 001 | 2 months | 0 | Male | 40 year | 1 |
| 002 | 3 months | 0 | Male | 30 year | 2 |
| 003 | 12 months | 1 | Female | 42 year | 1 |
Intuitively, I would model this using a Cox proportional hazard model, with the variable 'TimetoDisabilty' as the time until the occurrence of the disability, and 'Gender', 'AgeatDisability', and 'OccupationClass' as covariates. Monthly probabilities are derived from the survival function.
Now assume -because of practical/technical reasons- it is only possible to perform a GLM Binomial regression. I read that performing a GLM Binomial regression on data with pseudo observations is analogue to a Cox Discrete Time Survival model. To prepare the analysis, I transform the wide dataset to a long dataset (with pseudo observations, see, e.g., https://grodri.github.io/glms/notes/c7s6), in which each line is duplicated according the variable TimetoDisability. For example, the first line from the table above is transformed to 2 lines as is took 2 months to become disabled. The last line has a value 1 for the variable 'Disability', as the disability occured in month 2. The variable 'AgeatDisability' is transformed into the variable 'Age', now representing the age during that month. The right censored observation is transformed into 12 lines, all having the value zero for the variable 'Disability', as the disability is not observed. This becomes:
| PolicyNr | Duration | Disability | Gender | Age | OccupationClass |
| 001 | 1 | 0 | Male | 39 year 11 months | 1 |
| 001 | 2 | 1 | Male | 40 year | 1 |
| 002 | 1 | 0 | Male | 29 year 10 months | 2 |
| 002 | 2 | 0 | Male | 29 year 11 months | 2 |
| 002 | 3 | 1 | Male | 30 year | 2 |
| 003 | 1 | 0 | Female | 41 year 1 months | 1 |
| 003 | 2 | 0 | Female | 41 year 2 months | 1 |
| 003 | 3 | 0 | Female | 41 year 3 months | 1 |
| 003 | 4 | 0 | Female | 41 year 4 months | 1 |
| 003 | 5 | 0 | Female | 41 year 5 months | 1 |
| 003 | 6 | 0 | Female | 41 year 6 months | 1 |
| 003 | 7 | 0 | Female | 41 year 7 months | 1 |
| 003 | 8 | 0 | Female | 41 year 8 months | 1 |
| 003 | 9 | 0 | Female | 41 year 9 months | 1 |
| 003 | 10 | 0 | Female | 41 year 10 months | 1 |
| 003 | 11 | 0 | Female | 41 year 11 months | 1 |
| 003 | 12 | 0 | Female | 42 year | 1 |
Question:
In this long data format, the multiple rows (pseudo observations) for each person are not independent. We have repeated measures for each person.
However, I read in Therneau and Grambsch: (quote)
"One concern that often arises is that observations [on the same individual] are "correlated," and would thus not be handled by standard methods. This is not actually an issue. The internal computations for a Cox model have a term for each unique death or event time..."
So for a Cox Discrete Time Survival model, the dependency is not an issue.
However, I don't see how the dependency in the data is not an issue for a GLM Binomial regression?
Is it -given the dependency in the data- appropriate to perform a GLM to get trustworthy estimates of monthly probabilities? Or should I go for a mixed effect model?
Thank you.
